According to my knowledge so far, Discontinuous Galerkin Method can be applied under three themes. They are nodal based, modal based, and simply quadrature-free (doesn’t matter to classify it nodal or modal, since it does not require quadrature points).
Nodal based DG method is to approximate wave field (as well as medium properties) using Lagrange polynomials. The coefficient of the polynomial basis is the values of the wavefield at the given point. Usually Gauss-Lobatto-Legendre (GLL) points are used, since they can both used as interpolation points and quadrature points. In addition, Lebesgue constant is minimum at those points, or equivalently the Vandermonde matrix determinant is maximum. In 1D GLL points are also minimum potential electrostatic points, Fekete points. In 2D and up, nodal points are not easy to seek for simplex region (excluding quadrilateral, hexahedral regions). Most of the time the extended GLL points in multi-dimension are used as interpolation points as well as quadrature points same as 1D case. Nodal based DG is challenged in handling curve-linear elements.
On the other side, modal based DG is ready to allow for curve-sided elements and non-constant medium property. In addition, the polynomial basis order P and the quadrature order Q are adaptive to local elements. However, the computation demand is higher than nodal based DG.
I think (by Aug 23 2010), quadrature free DG is somewhere between nodal and modal based DG by restricting in straight-sided elements. Here are my reasons: (1) medium values within an element can be approximated by polynomial bases, so it allows for heterogeneous models. (2) based on this approximation, the numerical integral can thus be evaluated exactly beforehand and stored for later computation. It is equivalent to the numerical integrals in nodal and modal based DG. (3) the computation demands is not as high as modal DG, of course after scarifying the flexibility of curve-sided elements. But the local adaptivity can be implemented by choosing different degree of material polynomial order.
So now how important to have curve-sided elements in all modeling problems? Unclear so case by case. But tt is always good to double check the improvement after using curve-sided elements.